[1] Abbe, E. Beiträge zur theorie des mikroskops und der mikroskopischen wahrnehmung. Arch. f.ür. Mikrosk. Anat. 9, 413-418 (1873). doi: 10.1007/BF02956173
[2] Goodman, J. W. Introduction to Fourier Optics. 2nd edn. (Mcgraw-Hill, New York, 1996).
[3] Dürig, U., Pohl, D. W. & Rohner, F. Near-field optical-scanning microscopy. J. Appl. Phys. 59, 3318-3327 (1986). doi: 10.1063/1.336848
[4] Yang, H. et al. Super-resolution biological microscopy using virtual imaging by a microsphere nanoscope. Small 10, 1712-1718 (2014). doi: 10.1002/smll.201302942
[5] Upputuri, P. K. & Pramanik, M. Microsphere-aided optical microscopy and its applications for super-resolution imaging. Opt. Commun. 404, 32-41 (2017). doi: 10.1016/j.optcom.2017.05.049
[6] Fang, N. et al. Sub-diffraction-limited optical imaging with a silver superlens. Science 308, 534-537 (2005). doi: 10.1126/science.1108759
[7] Taubner, T. et al. Near-field microscopy through a SiC superlens. Science 313, 1595 (2006). doi: 10.1126/science.1131025
[8] Kehr, S. C. et al. Near-field examination of perovskite-based superlenses and superlens-enhanced probe-object coupling. Nat. Commun. 2, 249 (2011). doi: 10.1038/ncomms1249
[9] Jacob, Z., Alekseyev, L. V. & Narimanov, E. Optical hyperlens: far-field imaging beyond the diffraction limit. Opt. Express 14, 8247-8256 (2006). doi: 10.1364/OE.14.008247
[10] Salandrino, A. & Engheta, N. Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations. Phys. Rev. B 74, 075103 (2006). doi: 10.1103/PhysRevB.74.075103
[11] Liu, Z. W. et al. Far-field optical hyperlens magnifying sub-diffraction-limited objects. Science 315, 1686 (2007). doi: 10.1126/science.1137368
[12] Guerra, J. M. Super-resolution through illumination by diffraction-born evanescent waves. Appl. Phys. Lett. 66, 3555-3557 (1995). doi: 10.1063/1.113814
[13] Wei, F. F. & Liu, Z. W. Plasmonic structured illumination microscopy. Nano Lett. 10, 2531-2536 (2010). doi: 10.1021/nl1011068
[14] Liu, X. W. et al. Fluorescent nanowire ring illumination for wide-field far-field subdiffraction imaging. Phys. Rev. Lett. 118, 076101 (2017). doi: 10.1103/PhysRevLett.118.076101
[15] Gustafsson, M. G. L. Nonlinear structured-illumination microscopy: wide-field fluorescence imaging with theoretically unlimited resolution. Proc. Natl Acad. Sci. USA 102, 13081-13086 (2005). doi: 10.1073/pnas.0406877102
[16] Hell, S. W. Far-field optical nanoscopy. Science 316, 1153-1158 (2007). doi: 10.1126/science.1137395
[17] Rust, M. J., Bates, M. & Zhuang, X. W. Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM). Nat. Methods 3, 793-796 (2006). doi: 10.1038/nmeth929
[18] Betzig, E. et al. Imaging intracellular fluorescent proteins at nanometer resolution. Science 313, 1642-1645 (2006). doi: 10.1126/science.1127344
[19] Hess, S. T., Girirajan, T. P. K. & Mason, M. D. Ultra-high resolution imaging by fluorescence photoactivation localization microscopy. Biophys. J. 91, 4258-4272 (2006). doi: 10.1529/biophysj.106.091116
[20] Ayas, S. et al. Label-free nanometer-resolution imaging of biological architectures through surface enhanced raman scattering. Sci. Rep. 3, 2624 (2013). doi: 10.1038/srep02624
[21] Rivenson, Y. et al. Deep learning microscopy. Optica 4, 1437-1443 (2017). doi: 10.1364/OPTICA.4.001437
[22] Nehme, E. et al. Deep-STORM: super-resolution single-molecule microscopy by deep learning. Optica 5, 458-464 (2018). doi: 10.1364/OPTICA.5.000458
[23] Wang, H. D. et al. Deep learning enables cross-modality super-resolution in fluorescence microscopy. Nat. Methods 16, 103-110 (2019). doi: 10.1038/s41592-018-0239-0
[24] Barakat, R. Application of apodization to increase two-point resolution by the sparrow criterion. Ⅰ. Coherent illumination. J. Opt. Soc. Am. 52, 276-283 (1962). doi: 10.1364/JOSA.52.000276
[25] Barakat, R. & Levin, E. Application of apodization to increase two-point resolution by the sparrow criterion. Ⅱ. Incoherent illumination. J. Opt. Soc. Am. 53, 274-282 (1963). doi: 10.1364/JOSA.53.000274
[26] Ando, H. Phase-shifting apodizer of three or more portions. Jpn. J. Appl. Phys. 31, 557-567 (1992). doi: 10.1143/JJAP.31.557
[27] Boyer, G. R. Pupil filters for moderate superresolution. Appl. Opt. 15, 3089-3093 (1976). doi: 10.1364/AO.15.003089
[28] Boyer, G. & Sechaud, M. Superresolution by taylor filters. Appl. Opt. 12, 893-894 (1973). doi: 10.1364/AO.12.000893
[29] Boivin, R. & Boivin, A. Optimized amplitude filtering for superresolution over a restricted field Ⅰ. Achievement of maximum central irradiance under an energy constraint. Opt. Acta.: Int. J. Opt. 27, 587-610 (1980). doi: 10.1080/713820285
[30] Boivin, R. & Boivin, A. Optimized amplitude filtering for superresolution over a restricted field Ⅱ. Application of the impulse-generating filter. Opt. Acta.: Int. J. Opt. 27, 1641-1670 (1980). doi: 10.1080/713820181
[31] Boivin, R. & Boivin, A. Optimized amplitude filtering for superresolution over a restricted field Ⅲ. Effects due to variation of the field extent. Opt. Acta.: Int. J. Opt. 30, 681-688 (1983). doi: 10.1080/713821243
[32] Sales, T. R. M. & Morris, G. M. Fundamental limits of optical superresolution. Opt. Lett. 22, 582-584 (1997). doi: 10.1364/OL.22.000582
[33] Guillemin, E. A. The Mathematics of Circuit Analysis: Extensions to the Mathematical Training of Electrical Engineers (John Wiley & Sons, New York, 1949).
[34] Barnes, C. W. Object restoration in a diffraction-limited imaging system. J. Opt. Soc. Am. 56, 575-578 (1966). doi: 10.1364/JOSA.56.000575
[35] Frieden, B. R. On arbitrarily perfect imagery with a finite aperture. Opt. Acta.: Int. J. Opt. 16, 795-807 (1969). doi: 10.1080/713818225
[36] Di Francia, G. T. Super-gain antennas and optical resolving power. Il Nuovo Cim. 9, 426-438 (1952). doi: 10.1007/BF02903413
[37] Aharonov, Y., Albert, D. Z. & Vaidman, L. How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100. Phys. Rev. Lett. 60, 1351-1354 (1988). doi: 10.1103/PhysRevLett.60.1351
[38] Berry, M. V. Evanescent and real waves in quantum billiards and Gaussian beams. J. Phys. A: Math. Gen. 27, L391-L398 (1994). doi: 10.1088/0305-4470/27/11/008
[39] Berry, M. V. & Popescu, S. Evolution of quantum superoscillations and optical superresolution without evanescent waves. J. Phys. A: Math. Gen. 39, 6965-6977 (2006). doi: 10.1088/0305-4470/39/22/011
[40] Berry, M. V. & Dennis, M. R. Natural superoscillations in monochromatic waves in D dimensions. J. Phys. A: Math. Theor. 42, 022003 (2009). doi: 10.1088/1751-8113/42/2/022003
[41] Berry, M. V. & Shukla, P. Pointer supershifts and superoscillations in weak measurements. J. Phys. A: Math. Theor. 45, 015301 (2012). doi: 10.1088/1751-8113/45/1/015301
[42] Berry, M. V. A note on superoscillations associated with Bessel beams. J. Opt. 15, 044006 (2013). doi: 10.1088/2040-8978/15/4/044006
[43] Berry, M. V. Exact nonparaxial transmission of subwavelength detail using superoscillations. J. Phys. A: Math. Theor. 46, 205203 (2013). doi: 10.1088/1751-8113/46/20/205203
[44] Berry, M. V. & Moiseyev, N. Superoscillations and supershifts in phase space: wigner and Husimi function interpretations. J. Phys. A: Math. Theor. 47, 315203 (2014). doi: 10.1088/1751-8113/47/31/315203
[45] Berry, M. V. & Morley-Short, S. Representing fractals by superoscillations. J. Phys. A: Math. Theor. 50, 22LT01 (2017). doi: 10.1088/1751-8121/aa6fba
[46] Berry, M. V. Suppression of superoscillations by noise. J. Phys. A: Math. Theor. 50, 025003 (2017). doi: 10.1088/1751-8113/50/2/025003
[47] Berry, M. V. & Fishman, S. Escaping superoscillations. J. Phys. A: Math. Theor. 51, 025205 (2018). doi: 10.1088/1751-8121/aa9b50
[48] Liu, D. M. et al. Diffraction interference induced superfocusing in nonlinear Talbot effect. Sci. Rep. 4, 6134 (2014).
[49] Slepian, D. & Pollak, H. O. Prolate spheroidal wave functions, fourier analysis and uncertainty—Ⅰ. Bell Syst. Tech. J. 40, 43-63 (1961). doi: 10.1002/j.1538-7305.1961.tb03976.x
[50] Huang, F. M. & Zheludev, N. I. Super-resolution without evanescent waves. Nano Lett. 9, 1249-1254 (2009). doi: 10.1021/nl9002014
[51] Ferreira, P. J. S. G. & Kempf, A. Superoscillations: faster than the nyquist rate. IEEE Trans. Signal Process. 54, 3732-3740 (2006). doi: 10.1109/TSP.2006.877642
[52] Rogers, E. T. F. & Zheludev, N. I. Optical super-oscillations: sub-wavelength light focusing and super-resolution imaging. J. Opt. 15, 094008 (2013). doi: 10.1088/2040-8978/15/9/094008
[53] Wen, Z. Q. et al. Super-oscillation focusing lens based on continuous amplitude and binary phase modulation. Opt. Express 22, 22163-22171 (2014). doi: 10.1364/OE.22.022163
[54] Berry, M. V. Quantum backflow, negative kinetic energy, and optical retro-propagation. J. Phys. A: Math. Theor. 43, 415302 (2010). doi: 10.1088/1751-8113/43/41/415302
[55] Kempf, A. & Ferreira, P. J. S. G. Unusual properties of superoscillating particles. J. Phys. A: Math. Gen. 37, 12067-12076 (2004). doi: 10.1088/0305-4470/37/50/009
[56] Berry, M. V. Faster Than Fourier in Quantum Coherence and Reality. (World Scientific, Singapore, 1994).
[57] Yuan, G. H., Rogers, E. T. F. & Zheludev, N. I. "Plasmonics" in free space: observation of giant wavevectors, vortices, and energy backflow in superoscillatory optical fields. Light.: Sci. Appl. 8, 2 (2019). doi: 10.1038/s41377-018-0112-z
[58] Huang, K. et al. Optimization-free superoscillatory lens using phase and amplitude masks. Laser Photonics Rev. 8, 152-157 (2014). doi: 10.1002/lpor.201300123
[59] Zheludev, N. I. What diffraction limit? Nat. Mater. 7, 420-422 (2008). doi: 10.1038/nmat2163
[60] Huang, F. M. et al. Focusing of light by a nanohole array. Appl. Phys. Lett. 90, 091119 (2007). doi: 10.1063/1.2710775
[61] Wang, T. T. et al. Experimental verification of the far-field subwavelength focusing with multiple concentric nanorings. Appl. Phys. Lett. 97, 231105 (2010). doi: 10.1063/1.3524825
[62] Rogers, E. T. F. et al. A super-oscillatory lens optical microscope for subwavelength imaging. Nat. Mater. 11, 432-435 (2012). doi: 10.1038/nmat3280
[63] Li, M. Y. et al. Controllable design of super-oscillatory lenses with multiple sub-diffraction-limit foci. Sci. Rep. 7, 1335 (2017). doi: 10.1038/s41598-017-01492-y
[64] Grosjean, T. & Courjon, D. Polarization filtering induced by imaging systems: effect on image structure. Phys. Rev. E 67, 046611 (2003). doi: 10.1103/PhysRevE.67.046611
[65] Chen, G. et al. Super-oscillatory focusing of circularly polarized light by ultra-long focal length planar lens based on binary amplitude-phase modulation. Sci. Rep. 6, 29068 (2016). doi: 10.1038/srep29068
[66] Liu, T. et al. Subwavelength focusing by binary multi-annular plates: design theory and experiment. J. Opt. 17, 035610 (2015). doi: 10.1088/2040-8978/17/3/035610
[67] Wan, X. W., Shen, B. & Menon, R. Diffractive lens design for optimized focusing. J. Opt. Soc. Am. A 31, B27-B33 (2014). doi: 10.1364/JOSAA.31.000B27
[68] Chen, G. et al. Super-oscillation far-field focusing lens based on ultra-thin width-varied metallic slit array. IEEE Photonics Technol. Lett. 28, 335-338 (2016). doi: 10.1109/LPT.2015.2496148
[69] Chen, G. et al. Far-field sub-diffraction focusing lens based on binary amplitude-phase mask for linearly polarized light. Opt. Express 24, 11002-11008 (2016). doi: 10.1364/OE.24.011002
[70] He, Y. H. et al. Double-layer metallic holes lens based on continuous modulation of phase and amplitude. IEEE Photonics Technol. Lett. 26, 1801-1804 (2014). doi: 10.1109/LPT.2014.2333525
[71] Huang, K. et al. Ultrahigh-capacity non-periodic photon sieves operating in visible light. Nat. Commun. 6, 7059 (2015). doi: 10.1038/ncomms8059
[72] Dorn, R., Quabis, S. & Leuchs, G. Sharper focus for a radially polarized light beam. Phys. Rev. Lett. 91, 233901 (2003). doi: 10.1103/PhysRevLett.91.233901
[73] Hao, X. et al. Phase encoding for sharper focus of the azimuthally polarized beam. Opt. Lett. 35, 3928-3930 (2010). doi: 10.1364/OL.35.003928
[74] Kuga, T. et al. Novel optical trap of atoms with a doughnut beam. Phys. Rev. Lett. 78, 4713-4716 (1997). doi: 10.1103/PhysRevLett.78.4713
[75] Zhan, Q. W. Trapping metallic Rayleigh particles with radial polarization. Opt. Express 12, 3377-3382 (2004). doi: 10.1364/OPEX.12.003377
[76] Terakado, G., Watanabe, K. & Kano, H. Scanning confocal total internal reflection fluorescence microscopy by using radial polarization in the illumination system. Appl. Opt. 48, 1114-1118 (2009). doi: 10.1364/AO.48.001114
[77] Xue, Y. et al. Sharper fluorescent super-resolution spot generated by azimuthally polarized beam in STED microscopy. Opt. Express 20, 17653-17666 (2012). doi: 10.1364/OE.20.017653
[78] Hulteen, J. C. et al. Nanosphere lithography: size-tunable silver nanoparticle and surface cluster arrays. J. Phys. Chem. B 103, 3854-3863 (1999).
[79] Niziev, V. G. & Nesterov, A. V. Influence of beam polarization on laser cutting efficiency. J. Phys. D: Appl. Phys. 32, 1455-1461 (1999). doi: 10.1088/0022-3727/32/13/304
[80] Hafizi, B., Esarey, E. & Sprangle, P. Laser-driven acceleration with Bessel beams. Phys. Rev. E 55, 3539-3545 (1997).
[81] Quabis, S. et al. Focusing light to a tighter spot. Opt. Commun. 179, 1-7 (2000). doi: 10.1016/S0030-4018(99)00729-4
[82] Zhang, M. G. et al. Three-dimensional nanoscale far-field focusing of radially polarized light by scattering the SPPs with an annular groove. Opt. Express 18, 14664-14670 (2010). doi: 10.1364/OE.18.014664
[83] Venugopalan, P. et al. Focusing dual-wavelength surface plasmons to the same focal plane by a far-field plasmonic lens. Opt. Lett. 39, 5744-5747 (2014). doi: 10.1364/OL.39.005744
[84] Zakharian, A. R., Moloney, J. V. & Mansuripur, M. Surface plasmon polaritons on metallic surfaces. Opt. Express 15, 183-197 (2007). doi: 10.1364/OE.15.000183
[85] Liu, Y. X. et al. Far-field superfocusing with an optical fiber based surface plasmonic lens made of nanoscale concentric annular slits. Opt. Express 19, 20233-20243 (2011). doi: 10.1364/OE.19.020233
[86] Liu, T. et al. Vectorial design of super-oscillatory lens. Opt. Express 21, 15090-15101 (2013). doi: 10.1364/OE.21.015090
[87] Ye, H. P. et al. Creation of a longitudinally polarized subwavelength hotspot with an ultra-thin planar lens: vectorial Rayleigh-Sommerfeld method. Laser Phys. Lett. 10, 065004 (2013). doi: 10.1088/1612-2011/10/6/065004
[88] Yu, A. P. et al. Creation of Sub-diffraction longitudinally polarized spot by focusing radially polarized light with binary phase lens. Sci. Rep. 6, 38859 (2016). doi: 10.1038/srep38859
[89] Kozawa, Y. & Sato, S. Sharper focal spot formed by higher-order radially polarized laser beams. J. Opt. Soc. Am. A 24, 1793-1798 (2007).
[90] Kozawa, Y. & Sato, S. Focusing of higher-order radially polarized Laguerre-Gaussian beam. J. Opt. Soc. Am. A 29, 2439-2443 (2012). doi: 10.1364/JOSAA.29.002439
[91] Jiang, Y. S., Li, X. P. & Gu, M. Generation of sub-diffraction-limited pure longitudinal magnetization by the inverse Faraday effect by tightly focusing an azimuthally polarized vortex beam. Opt. Lett. 38, 2957-2960 (2013). doi: 10.1364/OL.38.002957
[92] Gu, Z. T. et al. Methods for generating a dark spot using phase and polarization modulation light. Optik 124, 650-654 (2013). doi: 10.1016/j.ijleo.2011.12.036
[93] Gan, Z. S. et al. Three-dimensional deep sub-diffraction optical beam lithography with 9nm feature size. Nat. Commun. 4, 2061 (2013). doi: 10.1038/ncomms3061
[94] Singh, R. K., Senthilkumaran, P. & Singh, K. Tight focusing of vortex beams in presence of primary astigmatism. J. Opt. Soc. Am. A 26, 576-588 (2009). doi: 10.1364/JOSAA.26.000576
[95] Chen, G. et al. Generation of a sub-diffraction hollow ring by shaping an azimuthally polarized wave. Sci. Rep. 6, 37776 (2016). doi: 10.1038/srep37776
[96] Wu, Z. X. et al. Binary-amplitude modulation based super-oscillatory focusing planar lens for azimuthally polarized wave. Opto-Electron. Eng. 45, 170660 (2018).
[97] Li, Z. Y. & Yu, N. F. Modulation of mid-infrared light using graphene-metal plasmonic antennas. Appl. Phys. Lett. 102, 131108 (2013). doi: 10.1063/1.4800931
[98] Yu, N. F. et al. Light propagation with phase discontinuities: generalized laws of reflection and refraction. Science 334, 333-337 (2011). doi: 10.1126/science.1210713
[99] Huang, L. L. et al. Dispersionless phase discontinuities for controlling light propagation. Nano Lett. 12, 5750-5755 (2012). doi: 10.1021/nl303031j
[100] Sun, S. L. et al. High-efficiency broadband anomalous reflection by gradient meta-surfaces. Nano Lett. 12, 6223-6229 (2012). doi: 10.1021/nl3032668
[101] Li, X. et al. Catenary nanostructures as compact Bessel beam generators. Sci. Rep. 6, 20524 (2016). doi: 10.1038/srep20524
[102] Yu, N. F. et al. A broadband, background-free quarter-wave plate based on plasmonic metasurfaces. Nano Lett. 12, 6328-6333 (2012). doi: 10.1021/nl303445u
[103] Zhao, Y. & Alù, A. Tailoring the dispersion of plasmonic nanorods to realize broadband optical meta-waveplates. Nano Lett. 13, 1086-1091 (2013). doi: 10.1021/nl304392b
[104] Luo, J. et al. Tight focusing of radially and azimuthally polarized light with plasmonic metalens. Opt. Commun. 356, 445-450 (2015). doi: 10.1016/j.optcom.2015.08.025
[105] Wang, S. Y. & Zhan, Q. W. Reflection type metasurface designed for high efficiency vectorial field generation. Sci. Rep. 6, 29626 (2016). doi: 10.1038/srep29626
[106] Li, Y. Y. et al. Broadband quarter-wave birefringent meta-mirrors for generating sub-diffraction vector fields. Opt. Lett. 44, 110-113 (2019). doi: 10.1364/OL.44.000110
[107] Zuo, R. Z. et al. Breaking the diffraction limit with radially polarized light based on dielectric metalenses. Adv. Opt. Mater. 6, 1800795 (2018). doi: 10.1002/adom.201800795
[108] McLeod, J. H. The axicon: a new type of optical element. J. Opt. Soc. Am. 44, 592-597 (1954). doi: 10.1364/JOSA.44.000592
[109] Hatakoshi, G. et al. Grating axicon for collimating Čerenkov radiation waves. Opt. Lett. 15, 1336-1338 (1990). doi: 10.1364/OL.15.001336
[110] García-Martínez, P. et al. Generation of bessel beam arrays through dammann gratings. Appl. Opt. 51, 1375-1381 (2012).
[111] Herman, R. M. & Wiggins, T. A. Production and uses of diffractionless beams. J. Opt. Soc. Am. A 8, 932-942 (1991). doi: 10.1364/JOSAA.8.000932
[112] Sabatyan, A. & Meshginqalam, B. Generation of annular beam by a novel class of Fresnel zone plate. Appl. Opt. 53, 5995-6000 (2014). doi: 10.1364/AO.53.005995
[113] Rogers, E. T. F. et al. Super-oscillatory optical needle. Appl. Phys. Lett. 102, 031108 (2013). doi: 10.1063/1.4774385
[114] Yuan, G. H. et al. Planar super-oscillatory lens for sub-diffraction optical needles at violet wavelengths. Sci. Rep. 4, 6333 (2014).
[115] Liu, T. et al. Focusing far-field nanoscale optical needles by planar nanostructured metasurfaces. Opt. Commun. 372, 118-122 (2016). doi: 10.1016/j.optcom.2016.04.022
[116] Qin, F. et al. Shaping a subwavelength needle with ultra-long focal length by focusing azimuthally polarized light. Sci. Rep. 5, 09977 (2015). doi: 10.1038/srep09977
[117] Ruan, D. S. et al. Realizing a terahertz far-field sub-diffraction optical needle with sub-wavelength concentric ring structure array. Appl. Opt. 57, 7905-7909 (2018). doi: 10.1364/AO.57.007905
[118] Wang, H. F. et al. Creation of a needle of longitudinally polarized light in vacuum using binary optics. Nat. Photonics 2, 501-505 (2008). doi: 10.1038/nphoton.2008.127
[119] Kitamura, K., Sakai, K. & Noda, S. Sub-wavelength focal spot with long depth of focus generated by radially polarized, narrow-width annular beam. Opt. Express 18, 4518-4525 (2010). doi: 10.1364/OE.18.004518
[120] Peng, R. B. et al. Super-resolution long-depth focusing by radially polarized light irradiation through plasmonic lens in optical meso-field. Plasmonics 9, 55-60 (2014). doi: 10.1007/s11468-013-9597-8
[121] Qin, F. et al. A supercritical lens optical label-free microscopy: sub-diffraction resolution and ultra-long working distance. Adv. Mater. 29, 1602721 (2017). doi: 10.1002/adma.201602721
[122] Yu, W. T. et al. Super-resolution deep imaging with hollow Bessel beam STED microscopy. Laser Photonics Rev. 10, 147-152 (2016). doi: 10.1002/lpor.201500151
[123] Lin, J. et al. Generation of hollow beam with radially polarized vortex beam and complex amplitude filter. J. Opt. Soc. Am. A 31, 1395-1400 (2014). doi: 10.1364/JOSAA.31.001395
[124] Chen, G. et al. Planar binary-phase lens for super-oscillatory optical hollow needles. Sci. Rep. 7, 4697 (2017). doi: 10.1038/s41598-017-05060-2
[125] Zhu, M. N., Cao, Q. & Gao, H. Creation of a 50, 000λ long needle-like field with 0.36λ width. J. Opt. Soc. Am. A 31, 500-504 (2014). doi: 10.1364/JOSAA.31.000500
[126] Dehez, H., April, A. & Piché, M. Needles of longitudinally polarized light: guidelines for minimum spot size and tunable axial extent. Opt. Express 20, 14891-14905 (2012). doi: 10.1364/OE.20.014891
[127] Khonina, S. N., Kazanskiy, N. L. & Volotovsky, S. G. Vortex phase transmission function as a factor to reduce the focal spot of high-aperture focusing system. J. Mod. Opt. 58, 748-760 (2011). doi: 10.1080/09500340.2011.568710
[128] Makris, K. G. & Psaltis, D. Superoscillatory diffraction-free beams. Opt. Lett. 36, 4335-4337 (2011). doi: 10.1364/OL.36.004335
[129] Zhang, S. et al. Synthesis of sub-diffraction quasi-non-diffracting beams by angular spectrum compression. Opt. Express 25, 27104-27118 (2017). doi: 10.1364/OE.25.027104
[130] Wu, Z. X. et al. Optimization-free approach for generating sub-diffraction quasi-non-diffracting beams. Opt. Express 26, 16585-16599 (2018). doi: 10.1364/OE.26.016585
[131] Greenfield, E. et al. Experimental generation of arbitrarily shaped diffractionless superoscillatory optical beams. Opt. Express 21, 13425-13435 (2013). doi: 10.1364/OE.21.013425
[132] Wu, J. et al. Creating a nondiffracting beam with sub-diffraction size by a phase spatial light modulator. Opt. Express 25, 6274-6282 (2017). doi: 10.1364/OE.25.006274
[133] Bokor, N. & Davidson, N. Generation of a hollow dark spherical spot by 4π focusing of a radially polarized Laguerre-Gaussian beam. Opt. Lett. 31, 149-151 (2006). doi: 10.1364/OL.31.000149
[134] Bokor, N. & Davidson, N. Tight parabolic dark spot with high numerical aperture focusing with a circular π phase plate. Opt. Commun. 270, 145-150 (2007). doi: 10.1016/j.optcom.2006.09.022
[135] Kozawa, Y. & Sato, S. Focusing property of a double-ring-shaped radially polarized beam. Opt. Lett. 31, 820-822 (2006). doi: 10.1364/OL.31.000820
[136] Xue, Y. et al. A method for generating a three-dimensional dark spot using a radially polarized beam. J. Opt. 13, 125704 (2011). doi: 10.1088/2040-8978/13/12/125704
[137] Li, S. et al. Generation of a 3D isotropic hollow focal spot for single-objective stimulated emission depletion microscopy. J. Opt. 14, 085704 (2012). doi: 10.1088/2040-8978/14/8/085704
[138] Wan, C. et al. Three-dimensinal visible-light capsule enclosing perfect supersized darkness via antiresolution. Laser Photonics Rev. 8, 743-749 (2014). doi: 10.1002/lpor.201400006
[139] Wu, Z. X. et al. Generating a three-dimensional hollow spot with sub-diffraction transverse size by a focused cylindrical vector wave. Opt. Express 26, 7866-7875 (2018). doi: 10.1364/OE.26.007866
[140] Tang, D. L. et al. Ultrabroadband superoscillatory lens composed by plasmonic metasurfaces for subdiffraction light focusing. Laser Photonics Rev. 9, 713-719 (2015). doi: 10.1002/lpor.201500182
[141] Yuan, G. H., Rogers, E. T. F. & Zheludev, N. I. Achromatic super-oscillatory lenses with sub-wavelength focusing. Light.: Sci. Appl. 6, e17036 (2017). doi: 10.1038/lsa.2017.36
[142] Khorasaninejad, M. et al. Achromatic metalens over 60nm bandwidth in the visible and metalens with reverse chromatic dispersion. Nano Lett. 17, 1819-1824 (2017). doi: 10.1021/acs.nanolett.6b05137
[143] Arbabi, E. et al. Controlling the sign of chromatic dispersion in diffractive optics with dielectric metasurfaces. Optica 4, 625-632 (2017). doi: 10.1364/OPTICA.4.000625
[144] Wang, S. M. et al. A broadband achromatic metalens in the visible. Nat. Nanotechnol. 13, 227-232 (2018). doi: 10.1038/s41565-017-0052-4
[145] Yuan, G. H. et al. Quantum super-oscillation of a single photon. Light.: Sci. Appl. 5, e16127 (2016). doi: 10.1038/lsa.2016.127
[146] Jin, N. B. & Rahmat-Samii, Y. Advances in particle swarm optimization for antenna designs: real-number, binary, single-objective and multiobjective implementations. IEEE Trans. Antennas Propag. 55, 556-567 (2007). doi: 10.1109/TAP.2007.891552
[147] Lin, J. et al. New hybrid genetic particle swarm optimization algorithm to design multi-zone binary filter. Opt. Express 24, 10748-10758 (2016). doi: 10.1364/OE.24.010748
[148] Li, W. L., Yu, Y. T. & Yuan, W. Z. Flexible focusing pattern realization of centimeter-scale planar super-oscillatory lenses in parallel fabrication. Nanoscale 11, 311-320 (2019). doi: 10.1039/C8NR07985D
[149] Li, J. L., Zhu, S. F. & Lu, B. D. The rigorous electromagnetic theory of the diffraction of vector beams by a circular aperture. Opt. Commun. 282, 4475-4480 (2009). doi: 10.1016/j.optcom.2009.08.028
[150] Carter, W. H. Electromagnetic field of a gaussian beam with an elliptical cross section. J. Opt. Soc. Am. 62, 1195-1201 (1972). doi: 10.1364/JOSA.62.001195
[151] Wolf, E. Electromagnetic diffraction in optical systems-Ⅰ. An integral representation of the image field. Proc. R. Soc. A 253, 349-357 (1959).
[152] Magni, V., Cerullo, G. & de Silvestri, S. High-accuracy fast Hankel transform for optical beam propagation. J. Opt. Soc. Am. A 9, 2031-2033 (1992). doi: 10.1364/JOSAA.9.002031
[153] Landau, H. J. & Pollak, H. O. Prolate spheroidal wave functions, Fourier analysis and uncertainty—Ⅱ. Bell Syst. Tech. J. 40, 65-84 (1961). doi: 10.1002/j.1538-7305.1961.tb03977.x
[154] Landau, H. J. & Pollak, H. O. Prolate spheroidal wave functions, Fourier analysis and uncertainty—Ⅲ: the dimension of the space of essentially time- and band-limited signals. Bell Syst. Tech. J. 41, 1295-1336 (1962). doi: 10.1002/j.1538-7305.1962.tb03279.x
[155] Slepian, D. Prolate spheroidal wave functions, Fourier analysis and uncertainty—Ⅳ: extensions to many dimensions; generalized prolate spheroidal functions. Bell Syst. Tech. J. 43, 3009-3057 (1964). doi: 10.1002/j.1538-7305.1964.tb01037.x
[156] Slepian, D. Prolate spheroidal wave functions, Fourier analysis, and uncertainty—Ⅴ: the discrete case. Bell Syst. Tech. J. 57, 1371-1430 (1978). doi: 10.1002/j.1538-7305.1978.tb02104.x
[157] Rogers, K. S. et al. Optimising superoscillatory spots for far-field super-resolution imaging. Opt. Express 26, 8095-8112 (2018). doi: 10.1364/OE.26.008095
[158] Karoui, A. & Moumni, T. Spectral analysis of the finite Hankel transform and circular prolate spheroidal wave functions. J. Comput. Appl. Math. 233, 315-333 (2009). doi: 10.1016/j.cam.2009.07.037
[159] Diao, J. S. et al. Controllable design of super-oscillatory planar lenses for sub-diffraction-limit optical needles. Opt. Express 24, 1924-1933 (2016). doi: 10.1364/OE.24.001924
[160] Yu, Y. Z. & Zhan, Q. W. Optimization-free optical focal field engineering through reversing the radiation pattern from a uniform line source. Opt. Express 23, 7527-7534 (2015). doi: 10.1364/OE.23.007527
[161] Liu, T., Yang, S. M. & Jiang, Z. D. Electromagnetic exploration of far-field super-focusing nanostructured metasurfaces. Opt. Express 24, 16297-16308 (2016). doi: 10.1364/OE.24.016297
[162] Khosrofian, J. M. & Garetz, B. A. Measurement of a Gaussian laser beam diameter through the direct inversion of knife-edge data. Appl. Opt. 22, 3406-3410 (1983). doi: 10.1364/AO.22.003406
[163] Born, M. & Wolf, E. Principles of Optics (Cambridge University Press, New York, 1999).
[164] Pernick, B. J. Two-dimensional light-distribution measurement with a 90° cornered knife edge. Appl. Opt. 32, 3610-3613 (1993). doi: 10.1364/AO.32.003610
[165] Xie, X. S. et al. Three-dimensional measurement of a tightly focused laser beam. AIP Adv. 3, 022110 (2013). doi: 10.1063/1.4791764
[166] Yang, L. X. et al. Minimized spot of annular radially polarized focusing beam. Opt. Lett. 38, 1331-1333 (2013). doi: 10.1364/OL.38.001331
[167] Huang, F. M. et al. Nanohole array as a lens. Nano Lett. 8, 2469-2472 (2008). doi: 10.1021/nl801476v
[168] Roy, T. et al. Point spread function of the optical needle super-oscillatory lens. Appl. Phys. Lett. 104, 231109 (2014). doi: 10.1063/1.4882246
[169] Wang, C. T. et al. Super-resolution optical telescopes with local light diffraction shrinkage. Sci. Rep. 5, 18485 (2015). doi: 10.1038/srep18485
[170] Wong, A. M. H. & Eleftheriades, G. V. Superoscillations without sidebands: power-efficient sub-diffraction imaging with propagating waves. Sci. Rep. 5, 08449 (2015). doi: 10.1038/srep08449
[171] Dong, X. H. et al. Superresolution far-field imaging of complex objects using reduced superoscillating ripples. Optica 4, 1126-1133 (2017). doi: 10.1364/OPTICA.4.001126
[172] Li, Z. et al. Achromatic broadband super-resolution imaging by super-oscillatory metasurface. Laser Photonics Rev. 12, 1800064 (2018). doi: 10.1002/lpor.201800064
[173] Fahrbach, F. O., Simon, P. & Rohrbach, A. Microscopy with self-reconstructing beams. Nat. Photonics 4, 780-785 (2010). doi: 10.1038/nphoton.2010.204
[174] Wong, A. M. H. & Eleftheriades, G. V. An optical super-microscope for far-field, real-time imaging beyond the diffraction limit. Sci. Rep. 3, 01715 (2013). doi: 10.1038/srep01715
[175] Matsunaga, D., Kozawa, Y. & Sato, S. Super-oscillation by higher-order radially polarized Laguerre-Gaussian beams. Proceedings of 2016 Conference on Lasers and Electro-Optics. (IEEE, San Jose, 2016).
[176] Yuan, G. H. et al. Flat super-oscillatory lens for heat-assisted magnetic recording with sub-50nm resolution. Opt. Express 22, 6428-6437 (2014). doi: 10.1364/OE.22.006428
[177] Eliezer, Y. et al. Breaking the temporal resolution limit by superoscillating optical beats. Phys. Rev. Lett. 119, 043903 (2017). doi: 10.1103/PhysRevLett.119.043903
[178] Eliezer, Y. et al. Experimental realization of structured super-oscillatory pulses. Opt. Express 26, 4933-4941 (2018). doi: 10.1364/OE.26.004933
[179] Eliezer, Y. & Bahabad, A. Super defocusing of light by optical sub-oscillations. Optica 4, 440-446 (2017). doi: 10.1364/OPTICA.4.000440